Abstract: We present the analysis on the accuracy and consistence properties and derive error estimates for the spectral conservative method for the elastic and inelastic space homogeneous Boltzmann problem as developed by Gamba and Tharskabhushanam (JCP 2009). The method is based on a Fourier representation formula of the collisional operator and a Lagrangian optimization correction for conservation of mass, momentum and energy for both the elastic and inelastic collision case. We present a discussion of the L^{1}-L^{2} theory for the scheme in the elastic case. This analysis allow us to obtain L^{2} and Sobolev error estimates for the approximating sequence securing the convergence properties of the scheme. The estimates are based on recent progress of convolution and gain of integrability estimates by the some of the authors and corresponding isomoment inequalities for the discretized collision operator.
Moreover we present recent developments of the space inhomogeneous simulations of boundary value problems for sudden cooling or heating phenomena where solutions to the non-linear Boltzmann equation is known to have discontinuities on velocity space near the boundary, and so strongly deviated from Maxwellian equilibrium.
This is work in collaboration with Ricardo J. Alonso and Sri Harsha Tharkabhushanam, and more recently with Jeff Haack. |